Second Order Elliptic Equations and Elliptic Systems

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Selected Title s i n Thi s Serie s 174 Ya-Zh e C h e n an d L a n - C h e n g W u , Secon d orde r ellipti c equation s an d ellipti c systems, 1 99 8 173 Y u . A . D a v y d o v , M . A . Lifshits , a n d N . V . S m o r o d i n a , Loca l propertie s o f distributions o f stochasti c functionals , 1 99 8 172 Ya . G . Berkovic h a n d E . M . Zhmud' , Character s o f finit e groups . Par t 1 , 1 99 8 171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 1 99 8 170 V i k t o r P r a s o l o v an d Yur i S o l o v y e v , Ellipti c function s an d ellipti c integrals , 1 99 7 169 S . K . G o d u n o v , Ordinar y differentia l equation s wit h constan t coefficient , 1 99 7 168 Junjir o N o g u c h i , Introductio n t o comple x analysis , 1 99 8 167 M a s a y a Y a m a g u t i , Masayosh i H a t a , an d J u n K i g a m i , Mathematic s o f fractals , 1 99 7 166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 1 99 7 165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n Galois theory , 1 99 7 164 E . I . G o r d o n , Nonstandar d method s i n commutativ e harmoni c analysis , 1 99 7 163 A . Ya . D o r o g o v t s e v , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko , Probability theory : Collectio n o f problems , 1 99 7 162 M . V . B o l d i n , G . I . Simonova , an d Yu . N . T y u r i n , Sign-base d method s i n linea r statistical models , 1 99 7 161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 1 99 7 160 V . G . Osmolovskff , Linea r an d nonlinea r perturbation s o f th e operato r div , 1 99 7 159 S . Ya . K h a v i n s o n , Bes t approximatio n b y linea r superposition s (approximat e nomography), 1 99 7 158 Hidek i Omori , Infinite-dimensiona l Li e groups , 1 99 7 157 V . B . Kolmanovski t an d L . E . Shatkhet , Contro l o f system s wit h aftereffect , 1 99 6 156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming , 1 99 7 155 Y u . Safaro v an d D . Vassiliev , Th e asymptoti c distributio n o f eigenvalue s o f partia l differential operators , 1 99 7 154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braid s an d 3-manifolds . A n introduction t o th e ne w invariant s i n low-dimensiona l topology , 1 99 7 153 S . K h . A r a n s o n , G . R . B e l i t s k y , an d E . V . Zhuzhoma , Introductio n t o th e qualitative theor y o f dynamica l system s o n surfaces , 1 99 6 152 R . S . Ismagilov , Representation s o f infinite-dimensiona l groups , 1 99 6 151 S . Yu . Slavyanov , Asymptoti c solution s o f th e one-dimensiona l Schrodinge r equation , 1996 150 B . Ya . Levin , Lecture s o n entir e functions , 1 99 6 149 Takash i Sakai , Riemannia n geometry , 1 99 6 148 V l a d i m i r I . P i t e r b a r g , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields, 1 99 6 147 S . G . Gindiki n a n d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s with quasihomogeneou s principa l part , 1 99 6 146 L . Ya . Adrianova , Introductio n t o linea r system s o f differentia l equations , 1 99 5 145 A . N . A n d r i a n o v an d V . G . Zhuravlev , Modula r form s an d Heck e operators , 1 99 5 144 O . V . Troshkin , Nontraditiona l method s i n mathematica l hydrodynamics , 1 99 5 143 V . A . M a l y s h e v an d R . A . M i n l o s , Linea r infinite-particl e operators , 1 99 5 142 N . V . Krylov , Introductio n t o th e theor y o f diffusio n processes , 1 99 5 141 A . A . D a v y d o v , Qualitativ e theor y o f contro l systems , 1 99 4 140 Aizi k I . Volpert , V i t a l y A . Volpert , an d Vladimi r A . Volpert , Travelin g wav e solutions o f paraboli c systems , 1 99 4 (Continued in the back of this publication)

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Second Orde r Elliptic Equation s and Ellipti c System s

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10.1090/mmono/174

Translations of

MATHEMATICAL MONOGRAPHS Volume 1 7 4

Second Orde r Elliptic Equation s and Ellipti c System s Ya-Zhe Che n Lan-Cheng Wu Translated b y BeiHu

•VttEHjj

American Mathematical Societ y Providence, Rhode Island Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Editorial Boar d Sun-Yung Alic e Chan g S.-Y. Chen g Tsit-Yuen La m (Chair ) Tai-Ping Li u Chung-Chun Yan g

(Second Orde r Ellipti c Equation s an d Ellipti c Systems ) Ya-Zhe Che n ( | ^ 5 E $ J T ) an d Lan-Chen g W u Copyright © 1 99 1 by Ya-Zhe Chen an d Lan-Chen g Wu Originally published i n Chinese by Science Press, Beijing, Chin a 1991 Translated fro m th e Chines e by Bei Hu ($ 3 IK) 1991 Mathematics Subject Classification. Primar y 35-01 , 35-02; Secondary 35B45, 35B65, 35J15, 35J60, 35K10. ABSTRACT. Thi s boo k i s base d o n th e authors ' lectur e note s a t th e Institut e o f Mathematic s a t Nankai Universit y durin g th e Partia l Differentia l Equation s Yea r i n 1 985 , absorbing als o th e mos t recent material s fro m th e lecture s o f experts . There ar e tw o part s o f th e book . Fo r th e Dirichle t proble m o f secon d orde r ellipti c partia l differential equations , variou s kind s o f a prior i estimat e methods , includin g th e mos t recen t tech niques, ar e rathe r completel y introduce d i n th e first part . Linear , quasilinea r an d full y nonlinea r equations ar e studied . I n th e secon d part , th e existenc e an d regularit y theorie s o f th e Dirichle t problem fo r linea r an d nonlinea r secon d orde r ellipti c partia l differentia l system s ar e introduced . This boo k choose s appropriat e materials ; i t i s a ver y goo d textboo k fo r graduat e students . This boo k ca n als o be use d a s a reference boo k fo r undergraduat e mathematic s majors , graduat e students, professor s an d scientists .

Library o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Chen, Yazhe , 1 939 [Erh chie h t' o yua n hsin g fan g ch'en g yi i t' o yua n hsin g fan g ch'en g tsu . English ] Second orde r ellipti c equation s an d ellipti c system s / Ya-Zh e Chen , Lan-Chen g W u ; translate d by Be i Hu . p. cm . — (Translation s o f mathematica l monographs , ISS N 0065-928 2 ; v. 1 74 ) Includes bibliographica l reference s an d index . ISBN 0-821 8-0970- 9 (alk . paper ) 1. Differentia l equations , Elliptic . I . Wu , Lancheng , 1 934 - . II . Title . III . Series . QA377.C44131 99 8 515'.353—dc21 97-4679 4 CIP

© 1 99 8 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . English translatio n publishe d b y th e AMS , wit h th e consen t o f Scienc e Press . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 0

9 08 07 06 05 04

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Contents Preface t o th e Englis h Translatio n x

i

Preface xii

i

Part I . Secon 1 d Orde r Ellipti c Equation s Chapter 1 . L 2 Theor y 3 1. Lax-Milgra m theore m 3 2. Wea k solution s o f ellipti c equation s 4 3. Th e Predhol m Alternativ e 7 4. A maximu m principl e fo r wea k solution s 8 1 5. Regularit y fo r wea k solution s Chapter 2 . Schaude r Theor y 1. Holde r space s 2. Mollifier s 2 3. C 2 ' a estimate s fo r solution s o f potentia l equation s 2 4. Interio r Schaude r estimate s 2 5. Globa l Schaude r estimate s 3 6. A maximu m principl e fo r classica l solution s 3 7. Solvabilit y o f th e Dirichle t proble m 3

3 7 7 0 3 7 0 2 3

Chapter 3 . L p Theor y 3 7 1. Th e Marcinkiewic z interpolatio n theore m 3 7 2. A decompositio n lemm a 4 0 3. Estimate s fo r solution s o f potentia l equation s 4 1 4. Interio r W 2iP estimates 4 6 5. Globa l W 2* estimate s 4 7 2 p 6. Existenc e o f W > solution s 4 9 Chapter 4 . D e Giorgi-Nash-Mose r Estimate s 5 1. Loca l propertie s o f wea k solution s 5 2. Interio r Holde r continuit y 6 3. Globa l Holde r continuit y 6

3 3 0 3

Chapter 5 . Quasilinea r Equation s o f Divergenc e For m 6 1. Boundednes s o f wea k solution s 6

7 7

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viii CONTENT

S

2. Holde r estimate s fo r bounde d wea k solution s 6 3. Gradien t estimate s 7 4. Gradien t Holde r estimate s 7 5. Solvabilit y o f th e Dirichle t proble m 7 Chapter 6 . Krylov-Safono v Estimate s 7 1. Th e Alexandroff-Bakelman-Pucc i maximu m principl e 7 2. Harnac k inequalitie s an d interio r Holde r estimate s 8 3. Globa l Holde r estimate s 9 Chapter 7 . Full y Nonlinea r Ellipti c Equation s 9 1. Maximu m nor m an d Holde r estimate 1 s fo r solution s 0 2. Gradien t estimate s 0 1 3. Gradien t Holde r estimate s 0 4. Solvabilit y fo r quasilinea r equation s o f nondivergenc 1 1 e for m 5. Solvabilit 1 y1 fo r full y nonlinea r equation s 1 1 6. A specia l clas s o f equation s 7. 1 Genera l full y nonlinea r equation s 2 Part II . Secon1 d Orde r Ellipti c System s 2

9 2 4 6 9 9 7 6 9 0 4 7 3 5 7 2 9

Chapter 8 . L 2 Theor y fo r Linea r Ellipti c System s o f Divergenc e 1 For m 3 1 1 1. Existenc e o f wea k solution s 3 1 2. Energ 1 y estimate s an d H 2 regularit y 3 4 Chapter 9 . Schaude r Theor y fo r Linea r Ellipti c System s o f Divergenc e For m 1 3 7 1 1. Morre y an d Campanat o space s 3 7 2. Schaude r theor y 4 5 Chapter 1 0 . LP Theory fo r Linea r Ellipti c System s o f Divergenc e For 1 m 5 1. BM O space s an d th e Stampacchi a interpolatio 1 n theore m 5 2. L p theor y 5

5 5 6

Chapter 1 1 . Existenc e o f Wea k Solution s o f Nonlinea r Ellipti c System 1 s 6 1. Introductio n 6 2. Th e variationa l metho d 6

3 3 4

Chapter 1 2 . Regularit y fo r Wea k Solution s o f Nonlinea r Ellipti c System 1 s 7 3 2 1. H regularit y 7 3 2. Furthe r regularit 1 y an d counterexample s 7 8 3. Indirec t metho 1 d fo r studyin g regularit y 8 1 4. Th e revers e Holde r inequalit y an d L p estimate 1 s fo r Du 8 7 5. Direc t method 1 s fo r studyin g regularit y 9 8 6. Th e singula r se t 20 4

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CONTENTS i

x

Appendix 1 . Sobole v Space s 20 1. Wea k derivative s an d th e Sobole v spac e W kiP(£l) 20 2. Rea l1 exponen t Sobole v space s H s(Rn) 2 3. Poincare' s inequalit y 2

9 9 2 3

Appendix 2 . Sard' s Theore m 2

5

Appendix 3 . Proo f o f the John-Nirenber 1 g Theore m 2

7

Appendix 4 . Proo f o f th e Stampacchi a Interpolatio1 n Theore m 2

9

Appendix 5 . Proo f o f the Revers e Holde r Inequalit y 22

5

Bibliographic Note s 23

3

Bibliography 23

9

Index 24

5

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Preface t o th e Englis h Translatio n We are very please d tha t ou r graduat e textboo k ha s bee n chose n b y the Amer ican Mathematica l Societ y fo r translatio n int o English . I n thi s Englis h edition , some obviou s typographi c error s i n th e Chines e editio n ar e corrected ; som e name s are modifie d accordin g t o th e curren t convention . W e als o adopted th e translator' s suggestion an d adde d ver y brie f bibliographi c note s fo r eac h chapter , togethe r wit h updated references . We would like to express our wholehearted gratitud e to Professors Alic e Chang, Tsit-Yuen La m an d othe r AM S personne l fo r thei r vas t amoun t o f wor k o n thi s translation project . W e would als o like to than k th e translator , Dr . Be i Hu, fo r hi s valuable suggestions ; h e type d th e WF^i. manuscrip t fo r thi s Englis h edition . Owing to the authors' limited knowledge , errors and inappropriatenes s ar e har d to avoid ; w e welcome correction s fro m readers . Ya-Zhe Chen , Lan-Chen g W u July 9 , 1 99 7

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Preface The theor y o f secon d orde r ellipti c equation s an d system s i s fundamenta l fo r studying partia l differentia l equations , an d therefor e i t wa s liste d a s a basic cours e for graduat e student s a t th e Institut e o f Mathematic s a t Nanka i Universit y durin g the Partia l Differentia l Equation s Yea r i n 1 985 . Th e author s wer e invite d t o giv e lectures t o graduat e student s fo r thi s course . A t tha t time , th e Institut e a t Nanka i also invited man y well-know n expert s fro m aroun d th e world to giv e lectures, which provided th e cours e wit h th e mos t recen t result s i n th e area . Thi s boo k i s base d on th e authors ' lectur e notes , but w e have absorbe d als o the mos t recen t material s from th e lecture s o f th e othe r invite d experts . There ar e man y excellen t book s o n secon d orde r ellipti c partia l differentia l equations an d systems , suc h a s [GT] , [LU ] and [GQl] , listed i n th e bibliograph y o f this book . Thes e book s giv e a thoroug h introductio n i n thi s area . However , som e of these books ar e too bi g to be suitable a s textbooks fo r beginners . Th e purpos e of this boo k i s to provid e a textboo k fo r graduat e students . Thi s boo k include s bot h basic material s an d th e mos t recen t result s an d methods , s o a s t o brin g graduat e students t o th e frontie r o f thi s area . There ar e tw o part s o f th e book . Fo r th e Dirichle t proble m fo r secon d orde r elliptic partia l differentia l equations , variou s kind s o f a prior i estimat e method s are rathe r completel y introduce d i n th e firs t part . Th e Krylov-Safono v estimat e and full y nonlinea r ellipti c equations , whic h appeare d i n th e 80's , ar e introduce d in detail , bu t concisely . I n th e secon d part , the existenc e an d regularit y theories o f linear an d nonlinea r secon d order ellipti c partial differential system s ar e introduced . Basic facts abou t Sobole v spaces are given in Appendix 1 . I n order t o emphasize th e main theme , th e proof s o f som e theorems , suc h a s th e Stampacchi a interpolatio n theorem an d th e revers e Holde r inequality , ar e include d i n th e Appendix . Owing to th e authors ' limite d knowledge , error s ar e har d t o avoid ; w e welcome suggestions fro m readers . Under th e leadershi p o f Professo r Li-Shan g Jian g (Hr^ L inj), th e partia l dif ferential equatio n semina r a t Pekin g Universit y playe d a n importan t rol e i n thi s book. W e would lik e to expres s ou r dee p gratitude t o Professo r Li-Shan g Jian g an d the semina r participant s fo r thei r contributions . W e would als o lik e to expres s ou r wholehearted gratitud e t o Professo r Guang-Li e Wan g (ZE^/fvl ) fro m Jili n Univer sity, wh o rea d th e draf t o f thi s boo k an d mad e valuabl e suggestions . Ya-Zhe Chen , Lan-Chen g W u May 20 , 1 99 0

xii i

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Part I

Second Orde r Ellipti c Equation s

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10.1090/mmono/174/01

CHAPTER 1

L2 Theor y The stud y o f solvability o f the Dirichle t proble m fo r ellipti c equations i s one of the central topics of this book. Th e introduction o f Sobolev spaces (cf . Appendi x 1 ) provides a n effectiv e too l fo r th e study . I n Sobole v spaces , w e ca n see k solution s in a mor e genera l clas s o f functions ; thi s make s solvabilit y problem s muc h easier . Solutions of this kind ar e often referre d t o as weak solutions o r generalized solutions. Of course, in order t o obtain th e existence of a classical solution, w e must stud y th e smoothness o f wea k solutions . Suc h a proble m i s called th e regularit y problem . I n §4 of this chapte r an d nex t chapter , w e will explore the basi c method s fo r studyin g the regularit y problem . 1. Lax-Milgra m theore m Let H b e a rea l Hilber t spac e an d H' it s dua l space . Denot e b y (• , •) the dua l product betwee n H an d H'. Definition 1 .1 . Le t a(u,v) b e a bilinea r for m o n th e Hilber t spac e H, (i) a(u, v) i s said t o b e bounded, if there exist s M > 0 such tha t (1.1) |a(u,t;)

| ^ M | M | H | M | H , VU,V

G H.

(ii) a(u,v) i s said t o b e coercive, if ther e exist s S > 0 such tha t (1.2) a(u,u)>5\\u\\

2

H,

VueH.

Theorem 1 . 1 (Lax-Milgra m theorem) . Let a(u,v) be a bounded, coercive bilinear form on H. Then for any f G H', there exists a unique u G H such that (1.3) a(u,v

) = (/,v) , VvGH,

and

(1-4) \\u\\

H.

Proof. Fo r eac h fixe d u G H, a(u, • ) is a bounde d linea r functiona l o n H, an d therefore ther e exist s a uniqu e Au G H' suc h tha t (1.5) a(u,v)

G H,

= (Au 9v), \fv

and \\Au\\w ^ M\\u\\

H.

3

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1. L 2 THEOR Y

4

It ca n easily b e verified tha t A i s linear. B y the coerciveness, w e have ||AU||/J'IMIH ^ > a(u,u) ^ S\\u\\jj, and therefor e (1.6) \\Au\\

H,

Z S\\u\\ H.

It follow s tha t A i s one to one. W e will prov e tha t th e range o f A i s R(A) = H'. First, w e show tha t R(A) i s closed. I f Au n— > v, then b y (1.6) \\un - u m\\H ^ ^\\Au n - Au

m\\H>\

thus {u n} i s a Cauchy sequence . Fro m th e completeness w e conclude tha t it s limit u E H exists . B y the continuit y o f A, w e derive tha t Au n —> Au, an d therefor e v = Au £ R(A). Thi s show s tha t R(A) i s a closed subspac e o f H'. I f R(A) ^ # ' , then i t follows from th e orthogonal decomposition theore m tha t ther e exists v' G H' such tha t v' ^ 0 and v' _ L R(A). Sinc e i 7 is reflexive, ther e exist s v G H suc h tha t (*>'» ')H' = (' v)> Usin g th e coerciveness, w e ge t 0 = (v',Av) Hf =

(Av,v) ^ S\\v\\

2 H

> 0.

This contradictio n implie s tha t R(A) = H\ an d therefore ther e i s a uniqu e u suc h that Au = f. Fro m (1 .5 ) and (1.6) we immediately deduc e (1 .3 ) and (1.4). D 2. Wea k solution s o f ellipti c equation s Let £1 b e an ope n domai n i n R n . Fo r simplicity, w e shall alway s assum e tha t n ^ 3 . In this chapte r w e shall conside r ellipti c equation s o f divergence for m o n ft: (2.1) Lu

= -Dj(a ijDiU +

d?u) + (VDiU + cu) = / + A f •

We us e the summation conventio n throughou t thi s book ; repeate d subscript s an d superscripts ar e summed fro m 1 through n , and Di = —— . A s for the operator L , OXi

we shall assum e throughou t thi s chapte r tha t aij eL°°(ft) , and tha t ther e exis t positiv e constant s A , A such tha t 2

(2.2) A|£|

^ ( x )^ « ; A|£|2, V£eR

n

,sefi,

nn

(2-3) £

iMU-cn ) + £ H i=l i=

rfi

H™ + IMIL""^) < A "

l

We shal l denot e th e Sobolev spac e W fc'2(ft) b y H k(tt). Fo r u,v G #H^)>

we s e t

a{u, v) = / | (a ljDiU + d ju)DjV + (b lDiU + cu) v [da;, t/ ri

From th e proof o f Lemma 2. 1 below w e shall see that th e integrals ar e well define d in the above equality .

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2. WEA K SOLUTION S O F ELLIPTI C EQUATION S 5

Definition 2.1 . Fo r T £ H~ l{Si) (th e dua l spac e o f H%{0,)) and g £ H we sa y tha t u £ H 1 (ft) i s a weak solution o f th e Dirichle t proble m

1

^),

f Lu = T i n ft, Iu = g o

n cK2,

if i t satisfie s

ju-seffoHfi) Lemma 2.1 . Le t £h e assumptions (2.2 ) anc i (2.3 ) 6 e in /orce, and let Q be an open bounded domain in R n . Then a(u,v) is a bounded bilinear form on HQ(Q). Proof. Usin g Holder' s inequalit y an d (2.2) , we obtai n / a lJDiuDjvdx ^ A | | u | | H i ( n ) | | t ; | | f f i ( n ) . I Jn Applying Holder' s inequality , (2.3 ) an d th e embeddin g theorem , w e deriv e

\i

djuDjvdx\ ^

di

2ll

l^ n ( n )IHlLa*(n)IPj«IUa(n)

3

< CMW\\ Hi(n)\MHl(n), I cuvdx I Jn

^ ll

C

ll L n / 2 (fi ) ll^l^ 2 * (O) IMIl*2* («)

< CA||u||

Hi(n)||t;||Hi(n),

where 2 * = 2n/(n — 2), an d th e constan t C depend s onl y o n n. Th e remainin g terms ca n b e estimate d i n a simila r way . Thu s w e obtai n

(2-6) K«,«)I^C'A||«||

if i (n) ||t;|| if i (n) .

•

Remark. Fo r eac h fixed u £ if 1 (Q), a(iz , •) i s a bounde d linea r functiona l o n HQ(Q). Analogousl y t o th e proo f o f Theore m 1 .1 , ther e exist s a bounde d linea r operator L : if1 (ft ) - * H~ 1 (fl) suc h tha t (2.7) a(u,v)

= (Lu,v), Vu£H

1

(n), v

£ H^(Q).

From no w o n w e shall no t distinguis h th e operato r L fro m th e operato r L give n i n (2.1). Lemma 2.2 . Let the assumptions (2.2 ) and (2.3 ) be in force, and let Q, be an open bounded domain in R n . Then there exists ~j2 > 0 such that a(u , v) +fi(u, v)^^ is coercive on HQ(Q) forfjL ^ Ji, where (•, -)L 2{SI) denotes the inner product on L 2(Q). In orde r t o prov e th e lemma , w e nee d th e followin g fact : fo r / £ L p(fL) an d arbitrary positiv e e > 0, ther e exist s a decompositio n f = f 1 + f 2 suc h tha t (2.8) Wf2\\

L'(Q)

< e, s*j>\h{x)\K,

where K i s sufficientl y large . Proof o f Lemm a 2.2 . Fo r an y e > 0 , ther e ar e decomposition s 6* =&*+&*, c f = d i + 4

>c

= ci+c 2,

such tha t

E U^H^ W + E I|4|IL"(Q ) + l|C2ll L»/a(n) ^ £>

E ii»iiu-(ii)+ E II4HL-(Q) + IMIL~(Q) < w . Set a2(u, v ) ~ \

(a ljD{U + d J2u)DjV + {b\DiU + C2ix) v [dx,

ai(ix,v) = a(u,v ) — a2(it,v) . Using th e positive-definitenes s conditio n (2.2 ) an d a simila r computatio n a s i n Lemma 2.1 , we easily obtai n a2(u,u) ^

(A-Ce)H^i

(n)

.

We fix e suc h tha t C e = A/4 . Fo r ai(tz,u) , w e hav e |ai(u,u)| ^

CTTO 0 such that for JJL > ~jl, the nonhomogeneous Dirichlet problem

{

Lu + uu = T , 1

u-^GfToHn)

/ms a unique weak solution.

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7

3. TH E FREDHOL M ALTERNATIV E

Proof. B y th e definitio n o f a wea k solution , th e bilinea r for m correspondin g to (2.9 ) i s give n b y a(u,v) + /i(ix , V)L 2 (Q)- A wea k solutio n satisfie s f a(u,v)+/x(u,v) o = /,

M + e + F0 - (k - I) We tak e fco— / = ( 1 — T))(M + e + Fo), wher e7 7 i s a small positiv e constan t t o be determined. The n

|A(* 0 )| 1 / 2 *^c[log-l . L^ J It i s clear tha t ther e exist s7 7 > 0 so tha t (4.1 5 ) i s valid. B y (4.1 7) , ess sup u ^ s u p u + + (1 - rj){M + e + F0 ) + CF 0 |ft| ( 1

/n)

~(1

/p)

.

Since M = ess sup u — sup u+ an d £ is arbitrary, w e conclude tha t dQ

esssupu ^ supu + + C F 0 | f i | ( 1 / n ) ~ ( 1 / p ) . The proo f i s complete . • Remark. I f the assumptio n (2.2 ) i s replaced b y th e stronge r assumptio n

(2-2)' £

IIVH^n ) + £ || n, then th e constan t C in the conclusio n (4.9 ) o f Theorem 4. 2 depend s only o n n,p , A/A an d ft an d i s independent o f 6*, ef, c and th e lowe r boun d fo r |H| .

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5. REGULARIT Y FO R WEA K SOLUTION S

13

In fact , fo r an y e > 0, w e can prov e tha t / \d?ipDjip + \Pu>Dj& + ap 2\dx\ 3 nI 3 J I

(4.24) U

For example , th e first ter m o n th e left-han d sid e o f (4.24 ) ca n b e estimate d a s follows: / d^ipDjipdx

^ II^IILHI^IIL^/(P-

I Jn

2 )II^^IU 2

Using (2.2) ' an d th e embeddin g theorem , w e then deriv e (4.24) . Thu s th e constan t C depend s onl y o n n,p , — an d Q. Ther e i s n o chang e i n othe r part s o f th e proof . A

The detail s ar e lef t a s exercise s fo r th e reader . Theorem 4.3 . Let the assumptions (2.1 ) , (2.2 ) be in force, and let c - Did 1 ^ 0 (in

the sense of £&'(Q)),

where Q is an open bounded domain on which the Sobolev embedding theorem is valid. Then there exists a unique weak solution to the Dirichlet problem, and (4-25) IMIffX(n

) < C(\\T\\ H-HQ) +

\\g\\

Hx(a)).

Proof. Withou t los s o f generality, w e may assum e tha t g = 0 (cf . th e proo f o f Theorem 2.3) . I f T = 0 , the zer o solution i s the onl y solution fo r (2.4) , by th e wea k maximum principl e (Theore m 4.1 ) . B y Theore m 3.2 , ther e exist s a uniqu e wea k solution u £ HQ(CI) fo r (2.4) , fo r an y T e H~ 1 (QI). I t follow s tha t L i s invertible , i.e., Nltfi(n) ^ C\\Lu\\ H-i(cl), Vu

e H^(Q).

From thi s inequality , (4.25 ) follows . D

5. Regularit y fo r wea k solution s For simplicity , w e consider onl y th e followin g typ e o f equation : (5.1) Lu

= -Dj(a ijDiu) +

VDiU + cu = f.

Theorem 5.1 . Let the assumption (2.2 ) be in force, and let a %i e W ltOC(Q), b\c e L°°(f2) , / G L2(ft). If u G if1 (ft ) is a weak solution of (5.1 ) , then for any ft' C C fi (i.e., ft' is a compact subset of SI) we have u E H 2(Qf), and (5-2) l|u||ff»2{fl). If a* J', 6 l, c and / ar e infinitely man y times differentiable, the n for an y k we have u G Wz * ' (ft) , an d therefore , b y th e Sobole v embeddin g theorem , u G C°°(ft) .

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10.1090/mmono/174/02

CHAPTER 2

Schauder Theor y The regularit y theor y fo r classica l solution s o f a secon d orde r ellipti c equa tion wa s firs t establishe d b y Schauder . A s fa r a s classica l solution s ar e concerned , the Schaude r theor y i s quit e complete . I t provide s a foundatio n fo r th e stud y o f nonlinear ellipti c equations . Here w e us e a ne w metho d introduce d b y Trudinger ; hi s proo f avoide d th e tedious computation s o f potentials .

1. Holde r space s In the process o f studying potentia l equations , it i s very inconvenient t o discus s continuous differentiabilit y alon e — th e result s ar e no t ric h an d plentiful . I t i s therefore necessar y t o introduc e th e concep t o f Holde r continuity , whic h ca n b e viewed a s fractiona l differentiatio n i n a certai n sense . Definition 1 .1 . Le t ft C M n, le t u(x) b e define d o n ft, an d le t XQ e f2 . If , fo r some 0 < a < 1 ,

then u i s said t o b e Holder continuous at xo with respect to Q, with exponent a. I f a — 1 in th e abov e definition , the n u i s said t o b e Lipschitz continuous at x$. We denote by C k{Sl) = C k,0(fl) th e space of functions whic h are /c-times continuously differentiat e o n Q. Fo r Holde r continuity , w e introduc e th e correspondin g class o f spaces, usuall y referre d t o a s Holde r spaces . For 0 < a ^ 1 , introduce th e semi-norm s (1.2) M o , 0 ;

« = [u] 0-Q = SU p \u(x)\,

(1.3) M o , a ;

Q = H a j Q = SU p H^

Q

(1.4) Mfc (1.5) Mfc.ajf

fo;n

Q[u]Q],

= M* ; n = X I I ^ o j f t * M=fc

i = Yl l D"u] 17

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2. S C H A U D E R T H E O R Y

18

where k i s a positiv e integer , v = {y\,v 2,- — ,^ n ) i s a multi-index , Vi ^ 0 (i = n

1,2,-•• ,n) , \v\ = ^2vi, an d 2 =1

D"u =

dx?~>dx%>

For simplicity , w e ofte n us e [D ku]a]n t o represen t V ^ [JD^UJQ-JQ . Definition 1 .2 . W e us e C fc ' a (Q) ( 0 < a ^ 1 ) t o denot e th e spac e consistin g of al l function s i n C k($t) satisfyin g [u]k^n < oo. Introduc e th e nor m o n C k(Q) k

(1.6) Mfc

;n=

5^M

TO;n ,

m=0

and th e nor m o n C fc ' a (Q) (1.7) |w|fc,a

;n

= Mfc ;n H - Mfc,a;n.

It i s no t har d t o verif y tha t bot h C k(Vt) an d C fc ' a (Q) ar e Banac h spaces . W e often dro p th e subscrip t Q in th e norm s i f ther e i s n o confusion . For th e produc t o f tw o functions , th e Holde r nor m satisfie s Lemma 1 .1 . For u,v G Ca (ft) ( 0 < a < 1 ) , (1.8) [uv]

a

^

[u] 0[v]a +

MaH o^

MaMa -

The proo f i s left t o th e reader . One o f th e mos t importan t propertie s i n Holde r space s i s th e interpolatio n inequality, whic h make s i t possibl e t o stud y onl y th e mos t importan t ter m i n de riving a n a prior i estimate , an d thu s simplifie s th e proof . Her e w e shal l us e th e compactness argumen t fo r it s proof . T h e o r e m 1 .2 . Let Q, be a bounded domain, and u € C Then for any e > 0, (1.9) [u]

2,a

(Ct) ( 0 < a ^ 1 ) .

2^e[u]2,a+C£\u\0,

(1.10) [u]i

^ e[u] 2,a + C e\u\o,

where C E depends on n , a, £1 , in addition to the dependence on e. All the norms and semi-norms are defined on Q. Proof. W e prov e onl y (1 .9) ; th e proo f fo r (1 .1 0 ) i s similar . I f ther e i s n o constant C e suc h that (1 .9 ) i s valid for al l functions i n C 2 , a (r2), then fo r an yA T > 0 , there exist s u^ suc h tha t (1.11) [u

N}2

> e[u N]2,a + N\u N\0 (N

= 1 ,2 , • • •) .

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1. H O L D E R SPACE S

19

Using th e homogeneit y o f th e inequality , w e can assum e withou t los s o f generalit y that \UN\2 = 1 , for otherwis e w e ca n replac e u^ b y UN/\UN\2- Fro m (1 .1 1 ) , (1.12) [u

N)2,a

< ~ , \u

N\o

= 1 ,2 , • • •).

< — {N

Since UN is uniforml y bounde d i n C 2 , a (Q), ther e exist s a subsequenc e u^ k whic h converges t o som e u i n C 2(S1 ), by th e Ascoli-Arzel a theorem . However , th e secon d inequality i n (1 .1 2 ) implie s tha t u^ converge s t o 0 uniformly o n Q . I t follow s tha t u = 0 , whic h contradict s \UN\2 — 1- Th e proo f o f (1 .9 ) i s complete . • Definition 1 .3 . A domai n f i i s sai d t o satisf y a cone property, i f ther e i s a finite con e V suc h tha t fo r an y x G fi, ther e exist s a con e congruen t t o V wit h vertex x an d containe d completel y i n Q,. Theorem 1 .3 . Suppose that Q satisfies a cone property with h the height of the cone. Then for any 0 < e ^ h, we have a

(1.13) [u]

(1.14) M i < e

2^e

[u}2}a +

1 +a

(j ^\u\

0j

M2,a + j H o ,

where the constant C depends only on n and the solid angle of opening of the cone. Proof. Le t V\ b e a con e wit h th e sam e soli d angl e o f openin g an d heigh t 1 . Then Theore m 1 . 2 implie s tha t (1.15) [ix]

2;vi

^ [ " h a ; * + C\u\ 0lVl fo

r u G C2 ' a (Fx),

where C depend s onl y o n th e soli d angl e o f opening . Fo r u G C2'a(VE) (V e share s the sam e soli d angl e o f openin g a s V\ , wit h heigh t e an d verte x a t th e origin) , w e use th e chang e o f variables y = x/e, u(y) = u{ey) = u(x). The n u G C2 ' a (Vi), an d (1.15) i s valid fo r u. I f w e rewrite thi s inequalit y i n term s o f the origina l variables , then Q M2;Ve ^ £ aM2,a;V£ + " ^ No-,Ve. For an y x G ft an d e ^ ft, ther e exist s a con e V e wit h verte x x an d V e C O . Thus C

C

Since a ; is arbitrary , M2;Q ^ f a M 2 , a ; f l + ~o Mofr-

The proo f o f (1 .1 4 ) i s similar. •

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20

2. SCHAUDE R THEOR Y

2. Mollifier s An equivalent Holde r nor m o f a function ca n b e introduced usin g derivative s of its mollifie d function . Thi s wil l reduce th e Holde r estimate s fo r solution s o f ellipti c equations t o estimatin g th e derivatives , an d therefor e wil l simplify th e proof . Let p £ C£°(R n ), p(x) ^ 0 with suppor t spt{p } C #i(0 ) an d (2.1) /

= 1.

p(x)dx

Such a functio n i s called a mollifier. On e ca n take , fo r example ,

{

k exp { —pr—- [ fo 0 fo

r \x\ < 1 ,

r \x\ ^ 0 ,

where k i s chosen s o tha t (2.1 ) i s valid . Definition 2 . 1 . Fo r u £ L/ oc (R n ) an d p{x) a mollifier , th e functio n

) = r~ n f p(^l)

(2.2) 5(x,r

u(y)dy

is said t o b e a mollified function o f u(x). Lemma 2.1 . Le i ^ £ C(W l). Then pact set to u(x) as r —» • 0 , an d (2.3) su

S(x,r ) converges uniformly on any com-

p |S| ^ su p |w|, k

(2.4) \D

k

u(x,r)\^Cr- su

p |u | (Jf

e = 0,1 ,2 , • • • ),

J3 T (x )

where D denotes the gradient in (x , r) an d C depends only on n, fc and the mollifier PProof. B y (2.1 ) , U(X,T)-U(X)

=

r~

n

/ p ( )(^(y)-w(a:))d2

jRn

(2.5) r =/

/

T

p(n)(u(x

- rrj) - u(x))drj,

^Bi(O)

from whic h i t i s no t difficul t t o deriv e tha t 5(x,r ) converge s uniforml y o n an y compact se t t o u{x) a s r - > 0 . T o prov e (2.4) , w e first notic e that , b y induction , Dku(x,T) =

r- n-k f

P

k(^^)u(y)dy,

where P k £ Co°(R n ), sp t P k C £?i(0) . Lettin g z = (x — y)/r i n th e abov e equality , we ge t \Dku(x,r)\ ^

r~

k

k(z)u(x

^ r~

k

su p \u\ / \P

\[ P

BT(x) JR

n

-

Tz)dz\ k{z)\dz

^

Cr~

k

su p \u\. D

B

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T{x)

2. MOLLIFIER S

21

Next w e shal l discus s th e relationshi p betwee n th e derivative s o f 5(x , r) an d the Holde r nor m o f u{x). Lemma 2.2 . Let u G C£c (R n ) ( 0 < a ^ 1 ) . Then ) - u{x)\ ^ r a # * [ u ; J5 r (z)],

(2.6) |5(s,r k

u(x, r)\ ^ CT"- kH«[w, B

(2.7) \D

T(x)]

k = 1 ,2 , • •. ,

for

where C depends only on n , a,A; and the mollifier p. Proof. Fro m (2.5 ) w e easil y deriv e (2.6) . T o prov e (2.7) , w e denot e b y / ? = (/30,/3) th e n + 1 dimensional multi-inde x wit h |/3 | = fc, D& = D!*>D%. I f / 3 = 0 , then w e differentiate (2.5 ) wit h respec t t o r t o obtai n D^(x,r) =

T - " - * J^{^-iy = r~

k

/ Pp(z)(u(x

u{y)-u{x))dy

- rz) - u(x))dx,

where Pp i s a functio n wit h it s suppor t i n B\(0). Fro difficult t o deriv e (2.7) . I f 0 ^ 0 , the n D?U(X,T)

= r~ n J n =

r

m thi s equalit y i t i s no t

D^U^)]u(y)dy

-£^[p(^)](u(y)-u(x))rfy + ( - l ) ^ l r - » u ( x ) f D*DP\p(Z^)\dy.

The secon d ter m o n th e right-han d sid e o f th e abov e formul a vanishes , b y th e divergence theorem . B y estimatin g th e first term , w e derive (2.7) . D Lemma 2.3 . Let u G C(Rn). If sup r

1

~a\Du(y,r)\
8)

P*C]

) = l forxGB

o + ( 1 - rr/T [I? f c C]a ^

(1

_^

Ti,, f c

,

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4. INTERIO R SCHAUDE R ESTIMATE S 2

9

where C — C(n,k), 0 < T < 1 . Suc h a functio n ca n be constructe d b y usin g a mollifier. Le t v = £u . The n v E C%,a(BR), an d Lv = C/ + [-a ijDi:iC + VDiQu - 2a

ij

Di(Dju.

In order to apply Lemma 4.2, we need to estimate the Holder norm of the right-han d side o f the above equation . W e firs t estimat e on e of the terms: A

[a^DiCDju]* ^

a {[DC]aNi

< r f [u]l

+ PC]oMi, a + [^C]oMi } fil

t

~alttM

^\ ( 1 - r) 1 +«fl 1 + t t ( l - r ) f l j ' Using th e interpolation inequalit y (Theore m 1 .3) , we obtain, fo r any £ G (0,1),

[««ACi>j«]. < cjel^uja + C ( 1 _ r ) ( i " i ; + 1 J ^ } . Similarly, b y Lemmas 4. 2 and 1.1, Theorem 1 . 3 and (4.8), [D2v]a,BR ^

(?

{_ +

^ [/]o;B

fl

+ [f]a;B

R

+

e[D

2

uUBR

(l_ r)(2/a)+li?2+alUlo^|-

Notice tha t £(x ) = 1 for x € B T R - Usin g th e product rul e fo r derivatives an d the interpolation inequalitie s (1 .1 3) , (1 -1 4) , we obtain {D2u]a.,BTR ^

2

cUf) + {1

a,BR+e[D

u]a,BR 1 fi(a/a)

- -ftNo.-B. + « ( 2 / Q ) + 1 - Q [ / ] O ; B B ] } •

-r)W*+W°»> [

Let dB+\dB! and |(/?iv — ^IO-B + "~ * 0 a

s i V -» oo .

The boundar y valu e proble m j -a

lj

DijUN +

[ UN — ^PN o

WDiUx + C-UJ V = / i

n B^",

n $£?+

2,a

has a solutio n ttj v G C (B1 ) , b y Theore m 7.1 . Usin g th e interio r Schaude r estimate an d th e maximum principle , w e see that a subsequenc e {?Mr fe} o f {u^r} converges uniforml y o n B^ t o som e functio n 2 , an d fo r an y Q' C C -B^ 1 ", {^jv fc} converges i n C 2(f2 ) to u. Lettin g N = iVf c—> o o in (7.7) , we obtain -aijDijU + u= uo

WDiU + cu = f i

n J5^,

n 95^".

It follow s tha t S = w in B*. Applyin g Lemm a 5. 2 to (7.7) , we ge t [D2uN]a;Bt/2